We covered this proposition during a lecture. However, I've been having a bit of trouble trying to prove it.
The definitions of the terms are given below
Here's how I'm thinking about it intuitively
Here's what I've got so far.
Any edge $e$ (that isn't a cut-edge) in $G$, separates two faces in $G$ (Don't know how to formally prove this. I think it has something to do with the Jordan Curve Theorem). Therefore, removing $e$, will reduce the number of faces by 1 in $G$. Therefore, two vertices in $G^*$ will be replaces by one vertex in $(G - e)^*$. This one vertex (that is mapped to the new face in $G - e$) will be connected to all the vertices that the two vertices were connected to (i.e. the new face will be connected to the faces that the previous faces were connected to). This is indicative of a contraction in $G^*$.
Obviously, this isn't a formal proof. I would appreciate some help in formalizing it.
I don't know at all what to do about the second equality.
I would appreciate the help